Methods and systems for component-based reduced order modeling for industrial-scale structural digital twins

ABSTRACT

A method for maintaining a physical asset based on recommendations generated by analyzing operational data and a composite model of a plurality of models representing the physical asset includes constructing, by a computing device, using a port-reduced static condensation reduced basis element approximation of at least a portion of a partial differential equation, the composite model. The computing device analyzes an error indicator associated with at least one model within the composite model to determine that the error indicator exceeds a tolerance level and increases a number of basis functions in the port-reduced static condensation reduced basis element approximation accordingly. The computing device receives first operational data associated with at least one region of the physical asset and updates the composite model. The computing device provides a recommendation for maintaining the physical asset, based upon the updated composite model.

BACKGROUND

The traditional approach to management of industrial machinery andinfrastructure from the point of view of structural integrity is toperform extensive analysis at design time to attempt to assess allrelevant operational conditions, and based on this analysis to decide on(i) the asset's operational life time, and (ii) a prescriptive scheme(often based on fixed time intervals) for inspection, maintenance, andrepair. A key observation in this work is that this design-basedprescriptive methodology is fundamentally limited by the large amount ofuncertainty about what the true operating conditions of the asset willbe. For example, consider the case of a seagoing vessel. The vessel isdesigned based on assumptions of the environmental conditions (waves,wind, corrosive seawater) and operating conditions (cargo, number andfrequency of loading/unloading cycles) to which it will be subjected.But, of course, the future operating conditions are unknown at designtime, so the only option is to make conservative assumptions and buildin large safety factors to compensate for uncertainty. In practice, whenthis type of design-time analysis is fully relied upon, this leads toover-design of assets (with corresponding excessive capital expenditure)or premature decommissioning compared to the true capacity of astructure or both. Moreover, even with conservative design assumptions,there is an ever-present risk of unforeseen circumstances duringoperations that go beyond the “worst case” assumed during design, suchas extreme weather, or accidents. Also, there is an increasing movementtowards lean design—especially in fields such as renewables where theeconomic viability of projects is often close to the break-even pointwhere safety margins are limited as much as possible in the interest ofreducing costs, which further increases the likelihood of an asset goingoutside its approved operating envelope. This brings health and safetyrisks and may also lead to damage that shortens asset lifetime ornecessitates expensive remedial interventions. Therefore, there is aneed for solutions that provide modeling of conditions andrecommendations for maintenance and safety throughout a physical asset'soperational lifetime.

BRIEF SUMMARY

In one aspect, a method for maintaining a physical asset based onrecommendations generated by analyzing a model of the physical asset,the model comprising a plurality of components and forming aphysics-based digital twin of the physical asset, includes constructing,by a computing device, using a port-reduced static condensation reducedbasis element approximation of at least a portion of a partialdifferential equation, a composite model of a plurality of models, eachof the plurality of models representing at least one of a plurality ofcomponents, each of the plurality of components representing at leastone region of a physical asset. The method includes analyzing, by thecomputing device, for at least one model in the plurality of models, anerror indicator identifying a level of error associated with the atleast one model, to determine that the identified level of error exceedsa tolerance level. The method includes increasing, by the computingdevice, a number of basis functions in the port-reduced staticcondensation reduced basis element approximation, based upon adetermination that the at least one model has a level of error exceedingthe tolerance level. The method includes repeating the error analysisand increasing of basis functions for each model in the plurality ofmodels until the level of error for each of the plurality of models isbeneath the tolerance level. The method includes receiving, by thecomputing device, from a first operational data source associated withthe physical asset, first operational data associated with at least oneregion of the physical asset represented by at least one parameter of atleast one component in the plurality of components. The method includesupdating, by the computing device, the composite model, based upon thereceived first operational data. The method includes providing, by thecomputing device, a recommendation for maintaining the physical asset,based upon the updated composite model.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, aspects, features, and advantages ofthe disclosure will become more apparent and better understood byreferring to the following description taken in conjunction with theaccompanying drawings, in which:

FIG. 1A is an illustration depicting how a system-level finite elementmodel may be obtained by connecting a plurality of components;

FIG. 1B depicts a hull model, updated based upon at least one valuewithin an inspection report;

FIG. 1C depicts updated hull models used to generate automated bucklingcheck reports;

FIG. 2A is a block diagram depicting an embodiment of a system formaintaining a physical asset based on recommendations generated byanalyzing a model of the physical asset, the model comprising aplurality of components and forming a physics-based digital twin of thephysical asset;

Referring now to FIG. 2B, a block diagram depicts a visualization of themethod 200 resulting in a digital thread for a floating offshorestructure.

Referring now to FIG. 2C, a block diagram depicts a visualization of themethod 200 resulting in a digital thread for an offshore platform.

FIG. 3 is a flow diagram depicting one embodiment of a method formaintaining a physical asset based on recommendations generated byanalyzing a model of the physical asset, the model comprising aplurality of components and forming a physics-based digital twin of thephysical asset; and

FIGS. 4A-4C are block diagrams depicting embodiments of computers usefulin connection with the methods and systems described herein.

DETAILED DESCRIPTION

The methods and systems described herein provide functionality formaintaining a physical asset based on recommendations generated byanalyzing a model of the physical asset, the model comprising aplurality of components and forming a physics-based digital twin of thephysical asset. In addition to maintaining the physical asset, themethods and systems described herein may provide functionality foridentifying one or more aspects of the physical asset that should beinspected (e.g., subject to a physical inspection). In addition tomaintaining the physical asset, the methods and systems described hereinmay provide functionality for determining a level of feasibility of aproposed modification to the physical asset. In addition to maintainingthe physical asset, the methods and systems described herein may providefunctionality for determining a level of operability of the physicalasset.

The methods and systems described herein may provide functionality forperforming structural integrity monitoring and reassessment duringoperation, which may enable identification of extra capacity present inthe asset (and hence may avoid early decommissioning) or overly onerousmaintenance regimes, while also tracking the impact of extreme orunpredictable events to ensure safety and reliability. In someembodiments, this goal of asset tracking and integrity monitoring duringoperations motivates the concept of a structural digital twin. The termdigital twin may refer to a computational replica of a physical assetwhich is kept in sync with the asset during its operational lifetime,based on inspection and sensor data, for example. A structural digitaltwin may refer to the specific case in which the purpose of the digitaltwin is to assess structural integrity based on the “as is” state of theasset. Updates to a structural digital twin may capture any structurallyrelevant changes to the asset, and can be based on, for example,inspection data (e.g. visual inspection, ultrasound thicknessmeasurements, laser scans) or sensor measurements (e.g. accelerometers,strain gauges, environmental monitoring). This inspection andinstrumentation significantly reduces, if not eliminates, theuncertainty associated with operating conditions since it allows forcontinuous updates the digital twin to reflect the true state of theasset and its environment. Through this approach, users of the methodsand systems described herein may therefore develop updated assetmanagement plans informed by the structural digital twin (e.g. forinspection, maintenance, repair, changes to allowable operatingconditions, damage or accident response, or asset life extension)instead of relying on the plans that were developed at design-time.Furthermore, it should be noted that structural digital twins may beprovided for physical assets within industrial systems, such as fixed orfloating offshore structures, aircraft, mining machinery, rotatingmachinery, or pressure vessels. The methods and systems describedherein, therefore, include a component-based reduced order modelingframework based on the Static Condensation Reduced Basis Element (SCRBE)method that enables fast, holistic, detailed, and parametric structuralanalysis of large-scale industrial systems. This methodology supportsmodeling needs of structural digital twins, where a structural digitaltwin is a detailed physics-based model of a structural system thattracks the “as is” state of the system over its operational lifetime. Arange of numerical examples will be discussed in further detail belowthat illustrate the unique capabilities of the SCRBE approach toincorporate inspection and/or sensor data, efficiently perform detailedstructural integrity analysis, and enable post-processing and reportgeneration to support data-driven decision-making during operation ofcritical structural systems.

The methods and systems described herein provide functionality forgenerating and updating structural digital twins that satisfy fourproperties: holistic and detailed modeling, speed, parametric modeling,and standards compliance and certifiable accuracy.

Structural digital twins should enable “screening” of an entire systemto identify the most likely “failure locations,” e.g. at stress andfatigue hotspots, so that these locations can be prioritized ininspection planning, for example. In order to enable this type ofscreening, a holistic and detailed model is used, in order to accuratelyresolve the stress throughout the components of the physical asset. Inorder to gain maximum insight from inspection data and sensor data, astructural digital twin should represent an entire asset as one holisticmodel, and should provide a sufficient level of detail such that allrelevant data can be mapped in an accurate manner into the digital twin.Through this approach the digital twin may capture the local, non-local,and cumulative effects of all updates (e.g. modifications, defects,damage) that have been measured or observed. This is an aspect of anassessment of the “as is” state of the asset that allows the system toensure that the accumulated effect of all of the updates to date doesnot lead to new and unexpected stress hotspots or predicted failuremodes. Holistic and detailed modeling is also desirable from the pointof view of workflow efficiency, in the sense that a model thatincorporates all up-to-date asset data in detail provides a “singlesource of truth” on the current state of the asset, and avoids the needto manage a multitude of separate localized models.

A status report from a structural digital twin typically involvessolving thousands of different load cases, e.g., to perform a fatiguelife estimation of critical parts, or to perform a strength check basedon the relevant industry standards under a wide range of “what if”scenarios. In order for an operator to be able to use the digital twinto inform data-driven decision-making, the full battery of analysisresults need to be completed quickly enough so that the status report isavailable in time for the decision-making process. Therefore, themethods and systems described herein provide functionality forgenerating and updating structural digital twins within a timeframerequired for the generated or updated structural digital twin to informdecision making.

An aspect of a structural digital twin is that it may evolve (and insome embodiments continuously evolves), either based on the updatedstate of the asset, or due to modifications imposed by the operator whomay want to assess different proposed changes or “what if” scenarios.Therefore, the methods and systems described herein providefunctionality for generating and updating structural digital twins thatare readily and efficiently modifiable. A parametric modeling approachmay facilitate such modifications, since parametric modeling enablesproperties to be modified via changing “dials,” e.g. to varystiffnesses, densities, loads, geometry, etc., and to be re-solvedautomatically and efficiently. Parametric modeling also facilitiesuncertainty quantification since model parameters can be statisticallysampled in order to assess uncertainty in the digital twin'spredictions, such as assessment structural fatigue under a range ofloading scenarios or corrosion rates. This type of statisticaluncertainty analysis allows for risk-based planning of asset managementbased on a digital twin.

Structural digital twins are typically deployed for safety critical andhigh-value assets, and there are many existing codes from regulators andstandards bodies that govern the type of analysis that should be used inthis context. Therefore, the methods and systems described hereinprovide functionality for generating and updating structural digitaltwins that comply with these standards in order for regulators andoperators to have full confidence in the results it provides whileproviding analysis results that may be checked to confirm their accuracyand reliability.

One conventional tool for structural integrity analysis of industrialequipment is the Finite Element (FE) method. FE certainly satisfiesrequirements around standards compliance and certifiable accuracy, butit has significant limitations for the other three items. Regardingholistic and detailed modeling and speed, the computational speed andmemory requirements of FE typically grow superlinearly with the numberof degrees of freedom, and hence in most practical circumstancesdetailed and holistic modeling is not feasible for large-scale systemswith FE. This issue with FE has led to the development of manysubmodeling-based workflows (coarse global models and separate finelocal models), but the submodeling approach ignores the non-local andcumulative effects we aim to capture when performing an update to astructural digital twin. Regarding parametric modeling, FE is notinherently parametric, in the sense that any parametric change requiresa new (often computationally intensive) solve to be performed fromscratch.

Artificial intelligence and machine learning (AI/ML), and relatedmethods such as response surfaces, are another set of candidatemethodologies that are often promoted for digital twins. AI/ML enablesfast analysis of systems, typically by evaluating specific quantities ofinterest (QoIs) as a function of parameters. As a result, AI/ML coversitems speed and parametric modeling well, but it falls short on theremaining two items. For holistic and detailed modeling, evaluation ofspecific quantities of interest is not consistent with the concept of aholistic and detailed model in which all details of an asset should befully represented, since the specific QoI outputs do not provide apicture of the entire asset. For standards compliance and certifiableaccuracy, AI/ML models are well-known to be “black boxes” which aredifficult to interpret, and which are not based on first principles ofphysics or compliant with physics-based asset integrity standards.

There is a wide range of reduced order modeling (ROM) methods that havebeen developed and which could be candidates for the type of structuraldigital twin discussed above, including Parabolic OrthogonalDecomposition (POD), Proper Generalized Decomposition (PGD), orCertified Reduced Basis Method; ROMs certainly provide speed,and—depending on the ROM type—may also provide parametric modeling andcertifiable accuracy. However, the ROMs generally do not enable holisticand detailed modeling of large-scale systems; therefore, they do nottypically provide a methodology that will apply to the largestindustrial systems, e.g. equivalent to more than 10⁸ FE degrees offreedom, as required for fully detailed models of large-scale floatingstructures, or aircraft, for example. The ROM approaches mentioned aboveare not well-suited to this type of large-scale model, since ROMs needto be “trained” by solving the full order model many times for differentconfigurations, and this is prohibitively expensive when the full-ordermodel is very large-scale.

Therefore, the methods and systems described herein provide acomponent-based ROM approach based on the Static Condensation ReducedBasis Element (SCRBE) framework. The SCRBE methodology builds on theCertified Reduced Basis Method to provide a physics-based ROM ofparametric partial differential equations (PDEs). As is typical for ROMapproaches, SCRBE involves an Offline/Online decomposition in which themodel data is “trained” during the Offline stage, and subsequentlyevaluated for specific parameter choices during the Online stage. TheOffline stage is computationally intensive, but once it is complete theOnline stage may be evaluated very quickly (typically orders ofmagnitude faster than a corresponding FE solve) for any new parameterchoices within a pre-defined range. The key aspect of the methodologythat differentiates it from the ROM methods discussed above, however, isthat it is component-based, in the sense that the overall system isdecomposed into smaller components and a separate ROM is trained foreach component. This enables greater scalability than other approachessince with SCRBE the system does not need to solve the entire systemwith a full order (e.g. FE) solver during the Offline stage—it issufficient to solve isolated components and local subsystems in order togenerate the training data. The resulting ROM for each componentconsists of reduced order representations of both the component interior(via the standard Reduced Basis method) and the component interfaces via“port reduction”. The baseline SCRBE method applies to linear PDEs sincethe formulation leverages static condensation, but it can be naturallyextended to incorporate nonlinearities by including nonlinear FE regionsin the model where needed. Such a SCRBE framework addresses all four ofthe properties of structural digital twins described above.

While the methods and systems described herein focus on structuraldigital twins based on the SCRBE methodology, in some embodiments, thesemethods and systems are combined with functionality for SCRBE, FE,and/or AI/ML. Therefore, the methods described herein may includereceiving output from an AI/ML system and incorporating that output intothe analyses; such methods may include providing output back to theAI/ML system with which the AI/ML system may automatically improve itssubsequent execution. In particular, AI/ML, may be highly effective as a“canary” in that it can generate a potential “red flag” quickly based onspecific QoIs during operations, and then SCRBE may be applied for afully-detailed analysis to assess the red flag scenario in more detailand prescribe further action if needed. Another combination of SCRBE andAI/ML uses SCRBE to generate physics-based data that can be used toaugment real-world measurements, and then the augmented datasets can beused to train a richer AI/ML model. This is particularly important inorder to enable an AI/ML model to accurately classify rare behavior(such as failures, which are typically rare on well-managed assets)since the real-world datasets on rare events is by definition limited.To address this issue, physics-based ROMs such as SCRBE can be used toefficiently generate a wide range of failure mode data by simulatingspecific failure scenarios and extracting virtual sensor readings inorder to augment and enrich AI/ML training sets. Similarly, SCRBE and FEcomplement each other well since SCRBE enables fast and parametricmodeling of large-scale systems, which can be used to identify localizedregions in the system which may have structural integrity issues. Once aregion is identified, it can be subjected to extensive localized FEanalysis to perform further assessment and since SCRBE is based on FEmeshes, the system may run FE using any subset of the components in anSCRBE model.

Although the structural digital twins considered in this work have SCRBEat their core, there is an extensive layer of pre-processing andpost-processing required in order to provide a complete workflow that isuseful to operators who manage critical systems on a day-to-day basis.Such a system should automatically and seamlessly update a structuraldigital twin based on new data, and generate reports that summarize keyfindings on demand. This flow from updated data to a digital twin to newreporting and recommendations is often referred to as a “digitalthread,” where the “thread” connects all relevant parts of the digitalasset integrity framework. Examples of a digital thread—built aroundSCRBE-based structural digital twins—for physical assets are describedin greater detail below.

Component-based modeling has long been a standard approach to analysisof large-scale structural systems. The starting point of acomponent-based formulation is typically to define a set of n_(comp)components, where component i corresponds to a spatial domain Ω_(i) andcontains a set of interface surfaces—that we refer to as ports—on whichcomponent i can be connected to neighbor components. Let n_(port) denotethe total number of connected ports in an overall system. Based on thiscomponent/port framework, the equivalent system-level FE model may beobtained by connecting the components to form the domain Ω=un_(i=1) ^(n)^(comp) Ω_(i), as illustrated in FIG. 1. As shown in FIG. 1, componentsfrom a hull (top) are assembled into a fully connected system-levelmodel (bottom). The SCRBE approach to developing parametric ROMs thatleverage the component-based decomposition of system-level modelsintroduced above will be described in greater detail below.

Let the system:KU=F,  (1)denote an equilibrium structural analysis FE problem (afterdiscretization based on a finite element space has been applied) posedon Ω, such as static or quasi-static linear elasticity. Here K∈

^(N) ^(FE) ^(×N) ^(FE) is the (symmetric)stiffness matrix, U∈

^(N) ^(FE) is the displacement vector, and F∈

^(N) ^(FE) is the load vector, where N_(FE) denotes the number of FEdegrees of freedom (DOFs) in the FE discretization on Ω. The system of(1) may be referred to as the standard FE formulation of this problem,and in the context of ROMs this is often referred to as the “truth”formulation.

Regarding the role of components, and for the sake of illustration, in ahighly simplified case with n_(comp)=2 and n_(port)=1, the system-leveldomain is Ω=Ω₁ ∪Ω₂, and p denotes the single port that connects Ω_(i)and Ω₂. Let N_(FE,1) and N_(FE,2) denote the number of FE DOFsassociated with the interior (non-port) region of components 1 and 2,respectively, and let N_(FE,p) denote the number of FE DOFs on μ. Notethat the DOFs on p may be standard FE Lagrange basis functionsassociated with individual nodes in the FE mesh, or they may be generalfunctions that have support on the entire port—the latter case isnecessary in the case of “port reduction,” which will be discuss ingreater detail below. Then (1) may be rewritten in the block form:

$\begin{matrix}{{\begin{bmatrix}K_{p,p} & K_{p,1} & K_{p,2} \\K_{p,1}^{T} & K_{1,1} & 0 \\K_{p,2}^{T} & 0 & K_{2,2}\end{bmatrix}\begin{bmatrix}U_{p} \\U_{1} \\U_{2}\end{bmatrix}} = {\begin{bmatrix}F_{p} \\F_{1} \\F_{2}\end{bmatrix}.}} & (2)\end{matrix}$

The matrix structure here suggests solving for U₁ and U₂ in terms ofU_(p) as follows:U _(i) =K _(i,i) ⁻¹(F _(i) −K _(p,i) ^(T) U _(p)),i=1,2.  (3)

Substitution of (3) into (2) then yields a system with only U_(p) asunknown:K _(p,p) U _(p)+Σ_(i=1) ² K _(p,i) K _(i,i) ⁻¹(F _(i) −K _(p,i) ^(T) U_(p))=F _(p),  (4)or equivalently(K _(p,i)−Σ_(i=1) ² K _(p,i) K _(i,i) ⁻¹ K _(p,i) ^(T))U _(p) =F_(p)−Σ_(i=1) ² K _(p,i) K _(i,i) ⁻¹ F _(i).  (5)The following notation may be used for the substructured stiffnessmatrix and load vector:

=(K _(p,p)−Σ_(i=1) ² K _(p,i) K _(i,i) ⁻¹ K _(p,i) ^(T)),

=F _(p)−Σ_(i=1) ² K _(p,i) K _(i,i) ⁻¹ F _(i),  (6)where

∈

^(N) ^(FE,p) ^(×N) ^(FE,p) , and

∈

^(N) ^(FE,p) Hence:

U _(p) =F  (7)

Here (7) is an exact reformulation of (1), where the key point is thatby performing a sequence of component-local solves as in (3) the systemis reduced to size N_(FE,p)×N_(FE,p) instead of the original sizeN_(FE)×N_(FE). This procedure is referred to by a number of differentnames in the literature, such as substructuring, superelements, staticcondensation or block Gaussian elimination. The name substructuring maybe used to refer to this approach.

The approach in (2)-(5) relies on the linearity of (1), and hence thecomponent-based formulation discussed in this section is linear-only.Approaches that apply to nonlinear analysis are discussed in greaterdetail below.

In order to implement (2)-(5) efficiently, in one embodiment, there isno explicitly computation of K_(i,i) ⁻¹. Instead, (3) is re-written as asequence of N_(FE,p)+1 solves—one for each DOF on port p plus one forF_(i)—as follows:K _(i,i) X _(i) ^(F) =F _(i) ,K _(i,i) X _(i) ^(j) =−K _(p,i) ^(T) e_(j) , j−1, . . . ,N _(FE,p),  (8)where e_(j) is the canonical unit vector in

^(N) ^(FE,i) with 1 in the jth entry, and once we have completed thissequence we can reconstruct K_(i,i) ⁻¹F_(i) and K_(i,i) ⁻¹K_(p,i) ^(T),and hence obtain the terms in (5) via pre-multiplication of K_(p,i). TheX vectors in (8) have no physical meaning, their role is only to be usedin the assembly of

and

.

In the derivation above we assumed that n_(comp)=2 and that there wasonly one port p, but everything generalizes naturally to the case of anynumber of components and ports. As a result, below we shall understandN_(FE,p) to refer to the number of FE DOFs on all ports in the system.

The substructuring approach described above is widely used, and affordssome attractive computational efficiencies and conveniences. Oneadvantage, as noted above, is that the system in (5) is typicallyconsiderably smaller than the system in (1). Another advantage is thatif a change is made within component i, in order to assemble the updatedN_(FE,p)×N_(FE,p) stiffness matrix and load vector we only need to solve(3) for component i rather than for all components, hence local changesto the component-based system can be performed in an efficientcomponent-local manner. While these advantages are attractive in somecases, in general the computational advantages of substructuringcompared to the standard FE formulation in (1) are quite limited due totwo key issues. First, the procedure described above for incorporating achange within a component is indeed modular, but it can be onerous sinceeach change to a component requires a new component-local FE solves tobe performed. In practice this may be costly, especially in the casethat many updates are required (e.g. when modifications are performed tomatch real-time sensor measurements, or within the inner loop of anoptimization), or when we use highly resolved component meshes. Second,the N_(FE,p)×N_(FE,p) matrix

is indeed smaller than (1) since the interior DOFs have been condensedout, but in general (in the case that n_(comp)>2) it is block-sparsewith potentially large and dense blocks with sizes corresponding to thenumber of port DOFs on component interfaces. Due to the extra density inthis matrix structure in many cases of interest it may require similaror even more computational resources to solve (7) than the originalsparse FE system (1). This is a well known issue with sub structuring,and the usual advice to address this is to make sure that ports containas few DOFs as possible (e.g. by locating ports in regions that aresmall, or coarsely meshed) to limit the size of the dense blocks. Inpractice these requirements impose very severe limitations on theapplication of sub structuring, and in many cases (depending on themodel geometry or mesh density) it is not possible for the requirementsto be satisfied.

The essence of the difficulty here is that substructuring is not a ROM.Instead, as noted above, it is an exact reformulation of the original FEproblem. This ensures the retention of full accuracy in the analysis,but on the other hand limitations (possibly severe) due to thecomputational issues noted above. The methods and systems describedherein, therefore, develop a ROM formulation that builds on thesubstructuring approach in order to address the computationallimitations of the standard substructuring approach. In particular thegoal here (as with any ROM approach) is to introduce a small extraapproximation compared to the non-reduced approach in return for a largecomputational advantage. In order to address the first issue listedabove, in one embodiment, we develop component-local parametric ROMsusing the Certified Reduced Basis (RB) Method in order to replace the FEsolves from (3) with parametrized RB solves. We introduce the parametervector μ∈

⊂

^(D), which encodes the parametrized properties in each component, suchas stiffnesses, shell thicknesses, densities, impedances, loads, orgeometry. The parameter domain

defines the min/max value of μ_(i), i=1, . . . ,D, and according to theRB framework this domain is set prior to performing the RB greedyalgorithm in the Offline stage since the RB greedy algorithm will sampleparameters within

. In all subsequent developments here and in the sections below we shallassume that the systems under consideration are parametrized by μ. Thenwe create an RB representation for each component so that we may replacethe component-interior FE solves—one solve per port DOF, as noted inremark 2.1)—with component-interior parametric RB solves. The first stepin this process is to introduce the affine expansion on component i asfollows:K _(i,i)(μ)=Σ_(q=1) ^(Q) ^(A) θ_(i) ^(K,q)(μ)K _(i,i) ^(q),(K_(p,i)(μ))^(T)=Σ_(q=1) ^(Q) ^(A) θ_(i) ^(K,q)(μ)(K _(p,i) ^(q))^(T) ,F_(i)(μ)=Σ_(q=1) ^(Q) ^(F) θ_(i) ^(F,q)(μ)F _(i) ^(q),  (9)where we separate into parameter-dependent functions (i.e. the θs) andparameter-independent operators (i.e. the K matrices and F vectors)—thisis crucial for the RB method's Online efficiency, as discussed below. Wecan then use (9) to reformulate the sequence of solves from (8) oncomponent i as follows:Σ_(q=1) ^(Q) ^(A) θ_(i) ^(K,q)(μ)K _(i,i) ^(q) X _(i) ^(F)(μ)=Σ_(q=1)^(Q) ^(F) θ_(i) ^(F,q)(μ)F _(i) ^(q),Σ_(q=1) ^(Q) ^(A) θ_(i) ^(K,q)(μ)K_(i,i) ^(q) X _(i) ^(j)(μ)=−Σ_(q=1) ^(Q) ^(A) θ_(i) ^(K,q)(μ)(K _(p,i)^(q))^(T) e _(j) , j=1, . . . ,N _(FE,p).  (10)

Next, we generate an “RB space” for each of the N_(FE,p)+1 parametricequations in (10). This is performed during the “Offline” stage using aRB greedy algorithm to generate a set of reduced basis functions thataccurately represent the full range of solutions for each of theequations in (10) over the entire parameter domain

. The RB greedy algorithm achieves this by using residual-based aposteriori error bounds in order to guide adaptive sampling in parameterspace in order generate efficient RB models that are accurate over theentire parameter domain of interest.

This results in N_(FE,p)+1 RB bases. In particular, let Z_(RB) ^(i)∈

^(N) ^(FE,i) ^(×N) ^(RB,i) denote an “RB basis function matrix” forcomponent i, where N_(RB,i) denotes the number of RB basis functions,and column j of Z^(i) is the jth basis function. We can then “reduce”the parameter-independent operators from (10) as follows:K _(i,i) ^(q,RB)=(Z _(RB) ^(i))^(T) K _(i,i) ^(q) Z _(RB) ^(i)∈

^(N) ^(RB,i) ^(×N) ^(RB,i) ,F _(i) ^(q,RB)=(Z _(RB) ^(i))^(T) F _(i)^(q)∈

^(N) ^(RB,i) ,   (11)and these reduced operators are stored to subsequently be used duringOnline solves. Typical sizes of N_(FE,i) and N_(RB,i) could be 0(10⁵)and 0(10), respectively, hence (11) represents a reduction from largesparse matrices to very small dense matrices, as is familiar from the RBframework.

In the “Online” stage, to assemble the contribution from eachcomponent's interior, the pre-stored reduced operators from (11) arecombined with the θs,—evaluated at the Online-requested parameter vectorμ—to assemble and solve the reduced system for any particular μ ofinterest. This assembly and solve is very fast since it depends only onquantities of size N_(RB,i), not N_(FE,i). This Online independence fromN_(FE,i) is referred to as the Offline/Online decomposition of the RBmethod, and it is enabled by the affine expansion from (10). In thecontext of SCRBE, the Offline/Online decomposition enables us to quicklyreassemble the system (5) after parametric properties within anycomponent (or all components simultaneously) are modified, whichdirectly addresses the first issue above.

The empirical interpolation method (EIM) can be applied in order toenable an approximate affine decomposition even in cases where an exactaffine decomposition is not available. This is often used in practice toenable certain types of complicated parametrizations including geometricmappings that “morph” a component's shape.

It is well known in the context of parametric ROMs in general, and theRB method in particular, that the Offline and Online computational costof ROMs generally increases rapidly as the number of parametersincreases—this is the so-called “curse of dimensionality.” However, theSCRBE framework circumvents this issue via component-based localizationof RB greedies and parameters: the RB greedy algorithm on component ionly involves the subset of parameters that affect component i. Thismeans we can set up large systems with many (e.g. thousands) ofparameters without being affected by the “curse of dimensionality” sincethe system-level model can be built from many components where eachcomponent only has a few parameters.

Regarding the second issue above, we note that the source of thecomputational difficulty is that conventional substructuring uses, inthe notation of (2), N_(FE,p) DOFs on the ports. The natural solution tothis issue is to reduce the number of DOFs on the ports by developingROMs for the port DOFs. This process of developing “port ROMs” isreferred to in the SCRBE literature as “port reduction” (In somearticles, “SCRBE with port reduction” is abbreviated as “PR-SCRBE,” butin this work we assume that we always apply port reduction, and hence weuse the shorter name “SCRBE” throughout.) We utilize various portreduction schemes from the literature, such as pairwise training andempirical modes, which involves port-based model reduction via properorthogonal decomposition (POD), or “optimal modes,” which solves atransfer eigenproblem to obtain an optimal set of port DOFs. The goal ofthese schemes is to construct a reduced set of N_(PR,p)(»N_(FE,p)) DOFson each port, while retaining accuracy compared to a full order solve byensuring that the dominant information transfer between adjacentcomponents is captured efficiently by the reduced set of port DOFs.Also, port reduction schemes operate based on small submodels of anoverall system during the Offline stage, so there is no need to performa full order system-level solve.

The reduced set of port modes means that instead of (7), we obtain asystem:

(μ)Ü _(p)(μ)=

(μ).  (12)of size N_(PR,p)×N_(PR,p), where

is also less dense than

due to the reduced size of the dense blocks. (Note that we also includethe parameter dependence in (12) now, due to the RB formulation oncomponent interiors, as introduced above.) Furthermore, port reductionreduces the Offline and Online cost associated with the RBapproximations on component interiors, since we replace N_(FE,p) in (10)with N_(PR,p).

Port reduction vastly increases the applicability of the substructuringframework since we may use ports of any shape or size and locate themanywhere in a model, and as long as we can generate an effective ROM forthe ports we will be able to solve the overall model efficiently. Inparticular, using SCRBE with port reduction for large-scale models wetypically obtain orders of magnitude speedup compared to a standard FEsolve, whereas with standard sub structuring in many cases the solvetime for the substructured system is comparable to, or even slower than,the solve time for the standard FE solve, as was noted above.

Perhaps more important than speedup compared to conventional FE, though,is the scalability that SCRBE with port reduction brings. Thecomputational cost of solving large-scale structural models with FE ofcourse grows as a function of N_(FE), where the growth rate depends onthe preconditioner type, solver type, and condition number of K. Inpractice most structural problems associated with industrial systemsinvolve ill-conditioning in one form or another, e.g. shell elements,slender solid elements, rigid connectors, or stiff beams, and thisill-conditioning means that it is highly unlikely that iterative solverswith preconditing such as the preconditioned conjugate gradient method,or GMRES, or algebraic multigrid methods, will converge. Sparse directsolvers are reliable solvers for FE formulations for the types ofproblems shown below. One advantage of a direct solver is that it avoidsany convergence issues, but one disadvantage is that it presentssignificant scalability issues for large-scale problems—especiallyassociated with memory requirements. In contrast, as described below,SCRBE with port reduction enables efficient solving of very large-scalemodels while avoiding convergence difficulties, since—as noted above—itdoes not require the performance of full order solves at the systemlevel during the Offline stage, and in the Online stage a reduced systemmay be constructed consisting only of port-reduced-DOFs that is smallenough to be solved quickly and efficiently with a direct solver. HenceSCRBE resolves a major computational limitation that is present withconventional FE solvers for large-scale structural problems. Tosummarize, therefore, the SCRBE framework presented here combines the RBmethod on component interiors and port reduction on component interfacesin order to resolve both the first and second issues described above.This enables fast, detailed, and parametric analysis of large-scalesystems, which is a set of capabilities that are ideally suited tostructural digital twins of industrial systems.

Another problem class of significant relevance to structural digitaltwins is frequency-domain analysis of forced vibration, such asHelmholtz acoustics or Helmholtz elastodynamics. In this case the FEsystem takes the form(K(ν)−ω² M(ν))U(ω,ν)=F(ω,ν),  (13)where now M denotes the FE-discretized mass matrix, ω is the frequency,and U is a complex-valued solution vector that represents, for example,the pressure (acoustics) or structural response (elastodynamics). In(13), let ν denote the user-specified parameter vector (e.g. materialproperties, geometry, loading, impedances), and then the full parametervector μ, as above, is given by μ=(ω, ν). Let H(μ)=K(ν)−ω² M(ν), so thatwe can rewrite (13) as:H(μ)U(μ)=F(μ).  (14)

One could apply the standard sub structuring framework (2)-(5) to (14)using the same approach as described above, and with the same drawbacksas identified in the first and second issues. The issue of extra densityin matrix structure is just as restrictive here as above, while theissue of cost when components change is typically much more restrictivein the frequency-domain case because it is very rare that an evaluationof (14) for a single frequency is sufficient—typically the goal of afrequency-domain analysis is to perform a “sweep” over a frequencyrange.

The SCRBE framework introduced above can once again be deployed toresolve these issues. This brings all the same benefits as above, andalso ensures that we can perform highly efficient frequency sweeps with(14) since the frequency ω is incorporated into the SCRBE parametric ROMin a natural way—it is simply treated as one of the entries of theparameter vector μ.

The SCRBE framework can be applied almost unchanged to (14). Onedifference is that now we develop an affine expansion of H(μ) instead ofK(μ), but this follows straightforwardly from the approach introducedabove, since the only extra requirement is that we must also create anaffine expansion of the mass matrix on each component, as follows:ω² M _(i,i)(ν)=Σ_(q=1) ^(Q) ^(M) θ_(i) ^(M,q)(μ)M _(i,i) ^(q) ,i=1, . .. ,n _(comp).  (15)Also, in order to ensure that all component interior solves from (3) arestable we must impose a limit on the parameter range of ω such thatω∈[0, λ_(comp)), where λ_(comp) denotes the smallest eigenvalue over allcomponents in the model of the generalized eigenvalue problem, i.e.λ_(comp)=min_(i=1) ^(n) ^(comp) min_(μ∈D)λ_(i) ¹(ν), whereK _(i,i)(ν)V _(i) ^(j)(ν)=λ_(i) ^(j)(ν)M _(i,i)(ν)V _(i) ^(j)(ν), j=1, .. . ,N _(FE,i),  (16)and where we assume that the eigenvalues are ordered such that λ_(i)¹(ν)≤λ_(i) ²(ν)≤ . . . ≤λ_(i) ^(N) ^(FE,) ^(i)(ν). This restrictionguarantees stability of the component-local solves because belowλ_(comp) the Helmholtz equation isolated to each component is coercive,whereas above λ_(min) we face instability as the inf-sup constant decaysto zero at resonances. It is important to note, however, that themaximum imposed on ω by λ_(comp) is usually not very restrictive in thecontext of system-level eigenvalues and eigenmodes, since typicallyindividual components are small compared to the overall system soλ_(comp) typically corresponds to a high-frequency at the system level.Once an SCRBE model is created and trained for a frequency-domainproblem, typically we can perform a fast “sweep” over a wide range offrequencies with dense sampling in ω in order to resolve complicatedvibrational responses with high accuracy.

Next, let us consider the parametrized dynamic model:M(μ)Ü(μ,t)+C(μ)Ü(μ,t)+K(μ)U(μ,t)=F(μ,t),  (17)which denotes the equation of motion of a structural system, after FEdiscretization has been applied spatially, where C is the dampingmatrix, and now the displacement field U(μ, t)∈

^(N) ^(FE) is a function of time. One can then discretize in time andapply standard explicit or implicit time-marching schemes in order tosolve this system, but this may be a highly computationally intensiveapproach, especially for large-scale systems. As a result, a commonalternative is to solve for the first N_(modes) eigenvalue/eigenmodepairs of the corresponding eigenvalue problem (where typicallyN_(modes)«N_(FE))K(μ)V ^(j)(μ)=λ^(j)(μ)M(μ)V ^(j)(μ), j=1, . . . ,N _(modes),  (18)and then we represent the dynamic solution via modal superposition onthe truncated set of eigenmodes:U(μ,t)=Σ_(j=1) ^(N) ^(modes) W _(j)(μ,t)V ^(j)(μ),  (19)where W(μ, t)∈

^(N) ^(modes) is the coefficient vector. Equivalently we can write U(μ,t)=ν(μ)W(μ, t), where ν(μ)∈

^(N) ^(FE) ^(×N) ^(modes) denotes the matrix in which column j is theeigenvector V^(j)(μ). We substitute (19) into (17) and applyorthogonality of eigenmodes with respect to M(μ) to obtain:{umlaut over (W)}(μ,t)+(ν(μ))^(T) C(μ)ν(μ){umlaut over(W)}(μ,t)+Λ(μ)W(μ,t)=(ν(μ))^(T) F(μ,t),  (20)where Λ(μ)=diag (λ₁(μ), . . . , λN_(modes)(μ)). (20) is a system ofN_(modes) ODEs that—if N_(modes) is large enough—generally captures theglobal dynamics of the system well.

However, if the structural system is large and/or complex, it may not becomputationally feasible to solve the global eigenproblem (18). As aresult, component-based model reduction frameworks have been extensivelydeveloped in order to enable efficient calculation of (18) and hence(20). The most widely known approach is the Craig-Bampton method, inwhich we form a reduced set of DOFs on each component based on (i)interface constraint modes, which are identical to the X_(i) ^(j) from(8), and (ii) a set of

fixed-interface normal modes, which are obtained via a component-localeigenproblem with zero constraints imposed on all ports. The combinationof (i) and (ii) corresponds to replacing U^(i) ∈

^(N) ^(FE,i) ^(+N) ^(FE,p) on component i with the “generalizedcoordinates” ξ^(i)∈

as follows:

$\begin{matrix}{{U^{i} = {\begin{bmatrix}{\Phi_{\ell}^{i}(\mu)} & {{- {K_{i,i}(\mu)}^{- 1}}( {K_{p,i}(\mu)} )^{T}} \\0 & I_{p,p}\end{bmatrix}\xi^{i}}},} & (21)\end{matrix}$where

is the matrix of the first

fixed-interface normal modes on component i, K_(i,i)(μ) and K_(p,i)(μ)are from (2), and I_(p,p)∈

^(N) ^(FE,p) ^(×N) ^(FE,p) is the identity matrix. This transformationcorresponds to a reduction in the number of interior DOFs on component idue to the truncation of the fixed-interface normal modes, since wetypically choose

«N_(FE,i). This DOF transformation can either be applied to (17)directly, or first to the modal problem (18) and then to the dynamicsystem, as in (20).

Many extensions have been proposed to the Craig-Bampton method, and theyare generally grouped into the family of Component Mode Synthesis (CMS)methods. CMS involves augmenting the Craig-Bampton representation withextra DOFs on the component interior. CMS methods (includingCraig-Bampton) are effective and widely used component-based ROMs sincethey capture the dominant modal and dynamic behavior of structuralsystems, while also providing a significant reduction in the number ofDOFs via the truncation of the component-interior DOFs. However, let usnow consider computational considerations for CMS, and in particular letus revisit the first and second issues above, in the context of CMSmethods.

First, regarding the same or additional computational resources requiredto solve (7) due to the extra density in the matrix structure, we notethat the standard formulation of CMS does not involve port reduction,i.e. as shown above the I_(p,p) block in (21) has sizeN_(FE,p)×N_(FE,p). However, approaches have been proposed to addressthis issue with CMS by reduction of the number of port DOFs usingtruncated eigenmodal representations on ports though the rather slowconvergence of eigenmodal expansions is a limitation of this approachand hence the pairwise training, empirical modes, or “optimal modes”approaches can provide an advantage over eigenmodal truncation in thiscontext. However, this issue is not addressed by CMS. It is clear fromthe dependence on μ in (17)-(21), that the quantities required toimplement the CMS formulation would have to be recalculated each time μis changed. Just as in the static, quasi-static, and Helmholtz cases,this is a major limitation of CMS for cases in which we wish to analyzea wide range of model configurations by changing parameters, e.g. in thecontext of design optimization, or for real-time model updates to matchsensor or inspection data in the context of structural digital twins.

To address the issue of computational cost when components change, acomponent-based parametric ROM eigensolver based on SCRBE has previouslybeen developed. The core idea of this SCRBE-based eigensolver is toreformulate the eigenproblem (18) to include the user-specifiedparameter vector μ as well as a shift parameter σ, such that:(K(μ)−σM(μ))V(μ)=τ(σ,μ)K(μ)V(μ),  (22)where τ is a shifted eigenvalue that satisfiesτ(μ^(i)(μ),μ)=0,  (23)where the λ^(i)(μ) are from (18). The next step is to develop an SCRBEapproximation for (22), which proceeds along the same lines as discussedfor Helmholtz problems in above, since the left-hand side matrix in (22)is exactly the H matrix from Section 2.2. Note that, as above, werestrict σ such that σ∈[0,λ_(comp)) in order to ensure coercivity andhence stability of the component interior solves for H, and once againthis is a modest restriction in practice since typically λL_(comp)corresponds to a high frequency at the system level.

Once the SCRBE-based ROM is constructed for (22), we can then use it inthe Online stage in order to assemble and solve a reduced eigenproblemfor any value of the pair (σ, μ). In practice, we use this capability asfollows: given a user-specified parameter vector μ, find values σ₀ ^(j)such that τ(σ₀ ^(j),μ)=0 for each j. We apply an iterative root-findingalgorithm, such as Brent's method, in order to find the σ₀ ^(j). Thesevalues then yield eigenvalues of the original system due to (23). Basedon this framework we are able to efficiently solve parametrizedeigenvalue problems via an SCRBE-based ROM for many different values ofμ and, following (20), the resulting eigenmodes can also be applied todynamic analysis of parametrized systems.

Next we consider application of component-based ROMs to nonlinearstructural analysis problems. One specific class nonlinearcomponent-based ROMs is flexible multi-body dynamics. This approachgeneralizes rigid multi-body dynamics, in which a system consists ofmultiple connected rigid components, and is widely used in industrialapplications, e.g. in modeling of drive-trains, and robotics. Inflexible multi-body dynamics, components in a rigid-body system may bereplaced by ROMs (typically using CMS) that represent the elasticresponse of the components. The overall analysis is then geometricallynonlinear due to the finite rotations and translations of each componentwithin the multi-body system, but each flexible component is assumed tohave a linear elastic response within its frame of reference. Flexiblemulti-body dynamics based on CMS is an effective approach within itsrealm of application, but that realm of application is quite specificand does not address the range of requirements that are needed forstructural digital twins, such as detailed stress and fatigue analysisincluding elastoplasticity, contact/friction, and large strain. As aresult, we focus on other approaches herein, but we note that rigid- orflexible-body dynamics approaches are a natural complement to the(linear or nonlinear) SCRBE-based model reduction discussed here becausethe multi-body dynamics analysis can provide loading data that can beimposed on the SCRBE-based model for detailed structural integrityanalysis.

Indeed, in general model reduction of nonlinear systems is a challengingproblem. Various so-called hyper-reduction strategies have beendeveloped to enable efficient nonlinear ROMs, such as empiricalinterpolation (EIM), discrete empirical interpolation (DEIM), or “gappyPOD” methods, which enable a full Offline/Online decomposition so thatthe Online ROM assembly and solve does not depend on N_(FE). Anotherapproach to nonlinear ROMs is machine learning (ML), in which we cannon-invasively train ML models based on supervised learning approaches,where a full order solver provides the “truth” data. But each of thesemethods inherently come with computational complexity or accuracylimitations so that in many cases the computational advantage of a ROMfor nonlinear problems may be debatable. This is especially true of“non-smooth” nonlinearities such as contact analysis andelastoplasticity, in which a small change in applied load can lead to adiscrete jump in the contact surface or “plastic front.” This type ofnon-smooth response is inherently difficult for ROMs to resolve, sinceROMs rely on construction of a low-dimensional representation of theresponse, and if the response is non-smooth an accurate low-dimensionalrepresentation may not exist (in mathematical terms, the Kolmogorovwidth of the response may be large, so that efficient model reduction isnot possible). There are certainly many valid and computationallyadvantageous methods for nonlinear ROMs which apply in specific cases,but in our view there is no single ROM approach that gives a significantcomputational advantage over full order models for the full range ofnonlinear analysis that is relevant for structural analysis, such ascontact/friction, elastoplasticity, and finite strain (e.g. forbuckling/post-buckling).

With the above considerations in mind, our approach to obtain a generalnonlinear solver that can be used for large-scale structural systems isto apply a so-called Hybrid Solver, which combines SCRBE in linearregions and FE in nonlinear regions within a fully-coupled global solve.We formulate this by first subdividing Ω into two subregions Ω_(lin) andΩ_(nonlin), where Ω_(nonlin) contains all nonlinearities and Ω_(1in)does not contain any nonlinearities.

On Ω_(lin) we apply the SCRBE framework from Section 2.1, which givesthe reduced system (12) of size N_(PR,p). On Ω_(nonlin) we introduce thenonlinear FE operator G_(FE,nonlin)(⋅;μ:

^(N) ^(FE,nonlin) →

^(N) ^(FE,nonlin) , where

^(N) ^(FE,nonlin) denotes the number of FE DOFs in Ω_(noniin). Let

$\begin{matrix}{{U_{hybrid}(\mu)} = \begin{bmatrix}{\hat{\mathbb{U}}(\mu)} \\{U_{{FE},{nonlin}}(\mu)}\end{bmatrix}} & (24)\end{matrix}$denote the global solution to the Hybrid SCRBE/FE system, where Û∈

^(N) ^(PR,p) is the solution on Ω_(lin), and U_(FE,nonlin) ∈

^(N) ^(FE,nonlin) is the solution on Ω_(nonlin). In general thisdefintion of U_(hybrid) permits discontinuity on the interface ofΩ_(nonlin) and Ω_(lin) so to enforce continuity we introduce aconstraint matrix C, which constrains the FE DOFs on the interface tomatch the SCRBE port modes such that CU_(hybrid) (μ) is continuous. Wemay then write the formulation on the entire domain Ω as:

$\begin{matrix}{{{G( {{U_{hybrid}(\mu)};\mu} )} = {C^{T}\begin{bmatrix}{{\hat{\mathbb{F}}(\mu)} - {{\hat{\mathbb{K}}(\mu)}{\hat{\mathbb{U}}(\mu)}}} \\{G_{{FE},{nonlin}}( { {{CU}_{hybrid}(\mu)} |_{\Omega_{nonlin}};\mu} )}\end{bmatrix}}},} & (25)\end{matrix}$where G(⋅; μ):

^(N) ^(R,p) ^(+N) ^(FE,nonlin) →

^(N) ^(R,p) ^(+N) ^(FE,nonlin) denotes the global nonlinear/linearoperator on Ω, and the C^(T) prefactor ensures that we use the same testfunctions as trial functions in the spirit of a Galerkin formulation.

We treat (25) as a nonlinear system with full two-way coupling betweenthe linear and nonlinear regions, and hence we solve it by applyingNewton's method to G. The Jacobian matrix J_(G)(μ)∈

^((N) ^(PR,p) ^(+N) ^(FE,nonlin)) ^(×(N) ^(PR,p) ^(+N) ^(FE,nonlin)) ofG is given by:

$\begin{matrix}{{{J_{G}( {{U_{hybrid}(\mu)};\mu} )} = {{C^{T}\begin{bmatrix}{- {\hat{\mathbb{K}}(\mu)}} & 0 \\0 & {J_{G_{{FE},{{non}lin}}}( { {U_{hybrid}(\mu)} |_{\Omega_{nonlin}};\mu} )}\end{bmatrix}}C}},} & (26)\end{matrix}$and then we apply the Newton iteration:J _(G)(U _(hybid) ^(k)(μ);μ)ΔU _(hybid) ^(k) =−G(U _(hybid)^(k)(μ);μ),  (27)U _(hybid) ^(k+1)(μ)=U _(hybid) ^(k)(μ)+ΔU _(hybid) ^(k),  (28)until we reach convergence.

Using formulation in (25) and (26) in implementation, we may assemblethe linear and nonlinear parts of the residual and Jacobianindependently based on the SCRBE formulation on Ω_(lin) and the FEformulation on Ω_(nonlin), and then the coupling of the two regions ishandled entirely by the matrix C.

In the context of structural digital twins, the Hybrid Solver approachhas several appealing features. First, it provides the full generalityof FE for accurately representing nonlinearities. Second, a digital twinmay require multiple separated nonlinear regions, e.g. due to damage orwear or failure in various separated parts of a large system. Due to thenature of the fully-coupled global nonlinear solve, the non-local andcumulative effects of all of these regions are automatically captured bythe Hybrid Solver. (In contrast, conventional “submodeling” workflowsignore non-local and cumulative effects.) Third, in the case that wehave linear predominance, the Hybrid Solver provides a largecomputational advantage compared to a full order solve, e.g. a speedupof 100× or more is typical; linear predominance refers to the case wherethe number of DOFs in the FE region is significantly smaller than thenumber of full order DOFs in the SCRBE region. We note that linearpredominance is common in structural digital twin applications, in whichoften nonlinearities are only required in regions with localized damageor failure, or localized contact regions, for example. In instances inwhich we do not have linear predominance, such as large deformationanalysis of an aircraft wing or wind turbine blade, and in which theentire model (or almost the entire model) must be treated as nonlinear,a global nonlinear FE solve, or, if applicable, one of the othernonlinear ROM methods cited above may be implemented.

Regarding error indicators and an adaptive ROM enrichment approach, aposteriori error assessment is an important ingredient in the SCRBEframework, for checking the accuracy of SCRBE solutions both during theOffline and Online stages. This provides guidance on when to haltOffline training, or when further ROM enrichment is required during theOnline stage. A posteriori error estimators for the SCRBE method (withport reduction) with respect to the “truth” FE formulation on Ω can bedeveloped. Error estimators for SCRBE have also been developed using aresidual-based approach, and, in order to make the approach rigorous, anumber of constants are computed in order to bound the error in terms ofthe residual (e.g. a stability factor for the operator, and constantsrequired to bound the dual norm of the residual, as is generallyrequired for residual-based error estimators). In the methods andsystems described below, we compute the residual with respect to the“truth” FE space. However, for the sake of simplicity, we omitcalculation of the extra constants required for detailed errorestimators, and instead use the residual directly as an a posteriorierror indicator. This residual-based error indicator approach fits wellin the context of structural digital twins, since the residual can beinterpreted as a “force balance” criterion, which for structuralengineers is a physically relevant quantity for indicating solutionaccuracy. Moreover, we compute the residual with respect to the discretefull order system and this quantity is typically used to determine astopping criterion in the context of iterative solvers (either linearKrylov subspace-type methods or nonlinear Newton-type methods), hencethe idea of assessing SCRBE solution accuracy based this quantity isnatural to engineers who are familiar with iterative solvers.

To make our formulation of the residual precise, we first must introduceÛ(μ), which is the SCRBE solution that is reconstructed on the entiresystem-level domain Ω based on a weighted sum of port DOFs and componentinterior DOFs scaled by coefficients from Ü_(p)(μ). Then we define theresidual

(μ) based on (1) as follows:

(μ)=F(μ)−K(μ)Ü(μ).  (29)

(μ) can be evaluated in a computationally efficient manner by treatingthe contribution to the residual from component interiors and portsseparately. Then finally we introduce our error indicator ε(μ)=∥

(μ)∥/∥ F(μ) ∥, which is the norm of the residual normalized by the normof the load.

We use the residual-based error indicator in both the Offline stage andthe Online stage, as described below. Note that in the description belowwe use the notion of a model, which refers to an assembly of SCRBEcomponents in which all parameters (materials, geometry, loads, etc.)are specified.

As has been noted, the use of the SCRBE framework in the methods andsystems described herein provides a powerful approach for enablingstructural digital twins of large-scale systems. These capabilities arefurther realized by connecting SCRBE-based models to inspection andsensor data and configuring post-processing for the purposes ofautomated asset integrity reporting. The data flow from operationalasset data (e.g., sensors or inspections), to structural digital twinupdate and analysis, to post-processing and reporting may be referred toas a “digital thread”. This digital thread may provide asset operatorswith deep structural integrity insights that leverage the asset data andthe SCRBE-based digital twin.

Referring now to FIG. 2A, in conjunction with FIG. 3, a method 200 formaintaining a physical asset based on recommendations generated byanalyzing a model of the physical asset, the model comprising aplurality of components and forming a physics-based digital twin of thephysical asset, includes constructing, by a computing device, using aport-reduced static condensation reduced basis element approximation ofat least a portion of a partial differential equation, a composite modelof a plurality of models, each of the plurality of models representingat least one of a plurality of components, each of the plurality ofcomponents representing at least one region of a physical asset (202).The method 200 includes analyzing, by the computing device, for at leastone model in the plurality of models, an error indicator identifying alevel of error associated with the at least one model, to determine thatthe identified level of error exceeds a tolerance level (204). Themethod 200 includes increasing, by the computing device, a number ofbasis functions in the port-reduced static condensation reduced basiselement approximation, based upon a determination that the at least onemodel has a level of error exceeding the tolerance level (206). Themethod 200 includes repeating, for each model in the plurality ofmodels, the analyzing of the error indicator and the increasing of thenumber of basis functions until the level of error for each of theplurality of models is beneath the tolerance level (208). The method 200includes receiving, by the computing device, from a first operationaldata source associated with the physical asset, first operational dataassociated with at least one region of the physical asset represented byat least one parameter of at least one component in the plurality ofcomponents (210). The method 200 includes updating, by the computingdevice, the composite model, based upon the received first operationaldata (212). The method 200 includes providing, by the computing device,a recommendation for maintaining the physical asset, based upon theupdated composite model (214).

The method 200, therefore, provides an Offline stage that executes anumber of steps. Method 200 includes specifying a set of trainingmodels,

, a training tolerance, TOL, and a number of training iterations,

. Method 200 includes, for each model M∈

, performing the following steps: (a) Solve with the current SCRBE ROMand calculate the error indicator ε(μ); and (b) If ε(μ)>TOL, performcomponent and port enrichment of the SCRBE ROM, as described above.Method 200 includes repeating steps (a) and (b)

times, or until ε≤TOL for all M∈

. The method 200 also includes an Online stage. In the Online stage, forany model M, the method 200 solves the system using the SCRBE ROM thatwas generated in the Offline stage. We may optionally also evaluate ε(μ)to validate the accuracy of the SCRBE solution, and if we find that E(μ)is larger than desired, we can run further enrichment by revisiting theprocedure from the Offline stage when needed—this can be performed in afully automated manner (driven by the error indicator), or the user maychoose to guide if and/or when further enrichment is to be performed.Thus, the availability of the error indicator provides a robust methodfor ensuring accuracy of our solutions in the Online stage. Also, if athorough Offline stage is performed, then typically we rarely need torevisit Offline calculations, and in most cases we will have a purelyOnline ROM that provides fast and accurate solutions for the full rangeof systems of interest. We refer to the Offline/Online proceduredescribe above as Adaptive ROM Enrichment (ARE).

Referring now to FIG. 2A, in greater detail and still in conjunctionwith FIG. 3, a method 200 for maintaining a physical asset based onrecommendations generated by analyzing a model of the physical asset,the model comprising a plurality of components and forming aphysics-based digital twin of the physical asset, includes constructing,by a computing device, using a port-reduced static condensation reducedbasis element approximation of at least a portion of a partialdifferential equation, a composite model of a plurality of models, eachof the plurality of models representing at least one of a plurality ofcomponents, each of the plurality of components representing at leastone region of a physical asset (202). The computing device 306 mayexecute an offline component 308 (which may be provided as either ahardware or a software component) that uses the SCRBE framework toconstruct the composite model of the plurality of models,

. As indicated above, in some embodiments, a hybrid solver is used toconstruct the composite model; in such embodiments, at least a firstportion of the partial differential equation is approximated using theSCRBE approach while FEA is applied to at least a second portion of thepartial differential equation.

The method 200 includes analyzing, by the computing device, for at leastone model in the plurality of models, an error indicator identifying alevel of error associated with the at least one model, to determine thatthe identified level of error exceeds a tolerance level (204). Theoffline component 308 may solve with the current SCRBE ROM and calculatethe error indicator.

The method 200 includes increasing, by the computing device, a number ofbasis functions in the port-reduced static condensation reduced basiselement approximation, based upon a determination that the at least onemodel has a level of error exceeding the tolerance level (206). If theerror indicator is greater than the tolerance level, then the offlinecomponent 308 enriches the SCRBE ROM (e.g., adds back in at least onebasis function), as described above. Basis functions (or degrees offreedom, as they are also described herein) may be added on bothinterfaces and interior of components. Component interior basisfunctions may be added following the reduced basis greedy algorithm, asdescribed above, in which residual-based a posteriori error bounds inorder to guide adaptive sampling in parameter space in order generateefficient RB models that are accurate over the entire parameter domainof interest. Component interface basis functions may be added based onadding data that captures the dominant information transfer betweenadjacent components

The method 200 includes repeating, for each model in the plurality ofmodels, the analyzing of the error indicator and the increasing of thenumber of basis functions until the level of error for each of theplurality of models is beneath the tolerance level (208). In oneembodiment, the method 200 includes repeating, for each model in theplurality of models, the analyzing of the error indicator and theincreasing of the number of basis functions until the level of error foreach of the plurality of models is beneath the tolerance level or untila threshold number of iterations is reached. As an example, the method200 may terminate (208) after determining that the tolerance level ismet by each model in the plurality of models. As another example, themethod 200 may terminate (208) after a predefined number of iterations;in this way, if each of the plurality of models cannot meet thetolerance level, the method 200 does not continue iterating endlessly.

In connection with the method 200, (202) (208) may be referred to as theoffline stage. In connection with the method 200, (202)-(208) may beperformed before the generation of a visual rendering of the compositemodel. In connection with the method 200, (202)-(208) may be performedbefore the receiving, by the computing device, from the firstoperational data source associated with the physical asset, firstoperational data associated with at least one region of the physicalasset represented by at least one parameter of at least one component inthe plurality of components.

In some embodiments, before generating an optional visual rendering ofthe composite model or receiving operational data, the method 200includes receiving user input (e.g., for modeling a “what if” scenario).Therefore, the method 200 may include receiving, by the computingdevice, first user input identifying an input value indicative of atleast one physical condition under which the physical asset is to beevaluated and using, by the computing device, the composite model togenerate at least one output value based at least in part on the atleast one input value, wherein the at least one output value isindicative of a behavior of the physical system under the at least onephysical condition, wherein the at least one output value comprises aplurality of output values over an N-dimensional domain. TheN-dimensional domain may be a 3-dimensional domain or any other value ofN. The user input may be any of a variety of input types. For example,the user input may identify an input value extracted from an inspectionreport based on a physical inspection of the physical asset. As anotherexample, the user input may identify an input value extracted fromoperational data received from a sensor associated with the physicalasset. As a further example, the user input may identify an input valuefor modeling a component under a particular operational condition, suchas to perform fatigue life estimation of critical parts or to perform astrength check based on an industry standard under at least oneoperational condition.

In some embodiments, the method 200 may include generating, by asimulation tool executed by the computing device, a visual rendering ofthe composite model including a visualization of at least one result ofa physics-based analysis of the physical asset. In some embodiments, thesimulation tool 304 generates the visual rendering of the compositemodel. The simulation tool 304 may generate a visual rendering of theentire composite model, including visualizations of all results of thephysics-based analysis of the physical asset. Alternatively, thesimulation tool 304 may visualization a subset of the resulting values;for example, the simulation tool 304 may visualization a level of stressat a single weld point as opposed to a level of stress throughout thephysical asset. The simulation tool 304 (which may be provided as ahardware component or as a software component) may generate the visualrendering. The simulation tool 304 may include a user interface withwhich a user of the system 300 may interact with the visual rendering ofthe composite model and provide user input. For instance, the userinterface 314 may allow the user to construct a model for a physicalsystem by specifying one or more aspects of the physical system, such asgeometry, material, and/or any other suitable physical characteristics.Once such a model is constructed, the user may, again via the userinterface 314, direct the simulation tool 304 to perform a simulationbased on the model to predict how the physical system may behave underone or more selected conditions. Results of the simulation may bedelivered to the user via the user interface 314 in any suitable manner,such as by visually rendering one or more output values of thesimulation. Therefore, an improved simulation tool is provided thatallows a user to modify one or more aspects of a physical system andobtain updated simulation results in real time. For instance, userinterface functions may be provided for the user to perform variousmodifications, including, but not limited to, modifying one or moreparameters of a component, adding a component (e.g., by cloning anexisting component), removing a component, disconnecting previouslyconnected components, moving a component from one part of the physicalsystem to another part of the physical system, and rotating a component.In response to such changes requested by the user, the simulation toolmay be able to quickly deliver updated simulation results by leveragingpreviously computed data. For example, in some implementations, thesimulation tool may update certain computations relating to componentswhose parameters and/or connections are changed but may reuse previouslycomputed data for components that are not directly affected by thechanges. In accordance with further embodiments, an improved simulationtool may perform one or more consistency checks to determine whetherchanges requested by a user are compatible with other aspects of aphysical system. The simulation tool may alert the user if anyincompatibility is detected. Additionally, or alternatively, thesimulation tool may propose further changes to the physical system toremove one or more incompatibilities introduced by the user-requestedchanges. In accordance with further embodiments, an improved simulationtool is provided that automatically computes an error associated with asimulation result. In some implementations, an error may be a rigorouslycomputed error bound, such as a maximum possible difference between thesimulation result and a result that would have been obtained had a fullFEA solution been computed. For example, in an embodiment in which RBapproximations are computed for component interior functions, a “local”error bound may be computed for each bubble function, where the localerror bound indicates a difference between a reduced order modelcomputed for the bubble function and a corresponding full FEA solution.Such local error bounds may then be combined to obtain an overall errorbound for an entire physical system or a portion thereof. In otherimplementations, an error may be an error estimate that can be computedin less time compared to a rigorous error bound. In yet some otherimplementations, a user may choose which type of error (e.g., rigorouserror bound or error estimate) is to be computed by the simulation tool.

The method 200 includes receiving, by the computing device, from a firstoperational data source associated with the physical asset, firstoperational data associated with at least one region of the physicalasset represented by at least one parameter of at least one component inthe plurality of components (210). The online component 310 executing onthe computing device 306 may receive the first operational data. Thecomputing device 306 may receive from the first operational data sourceassociated with the physical asset, first operational data generated bya sensor associated with the physical asset. The computing device 306may receive from the first operational data source associated with thephysical asset, first operational data extracted from an inspectionreport associated with the physical asset; for example, the inspectiondata may include a result from a visual inspection of an asset (e.g., anobservation that there is a crack or corrosion on a pipe and thatinformation should be built into the digital twin). The computing device306 may receive from the first operational data source associated withthe physical asset, first operational data extracted from a reportgenerated by an operator of the physical asset. The operational datainputs to a digital thread may include the inspection and sensor dataavailable from operational assets. Examples include, without limitation,thickness measurements based on ultrasound thickness gauging;environmental monitoring at specific intervals in time, e.g. wind andwave states for an offshore structure; operational load monitoring, e.g.throughput rates, tank fill levels, and number of loading/unloadingcycles per time interval; measurements from structural sensors such asaccelerometers and strain gauges; pressure and/or temperaturemonitoring.

The method 200 includes updating, by the computing device, the compositemodel, based upon the received first operational data (212). Thecomputing device may identify, within the received first operationaldata, an input value indicative of at least one physical condition underwhich the physical asset is to be evaluated and use the composite modelto generate at least one output value based at least in part on the atleast one input value, wherein the at least one output value isindicative of a behavior of the physical system under the at least onephysical condition, wherein the at least one output value comprises aplurality of output values over an N-dimensional domain. The computingdevice may execute an importer application that receives measurementdata, formats the data, and updates the model to incorporate measurementdata; for example, for thickness measurements, the computing device mayreceive measurement data in a document including the measurement data ina comma-separated values format (e.g., in a spreadsheet) and may updatethe SCRBE model's thickness to match the imported measurements. Forsensor data, the computing device may receive the data in an agreed-uponformat and incorporate the received data into the SCRBE model. Sensorreadings may be received in a text format and then importer software maybe configured to read the agreed-upon text format and apply the data tothe model. The computing device may receive an identification of wheresensors are installed on a physical asset. The computing device mayreceive an identification of a format for inspection data specifyingwhere in the physical asset each measurement is coming from. Theformatting may be agreed upon between the operator or owner of thephysical asset and the user generating the digital twin when the digitaltwin is being initially configured so that the digital twin will beconfigured in a manner consistent with the operational data that will bereceived. Once the computing device receives the operational data fromthe operational data source, whether an operator observation or a sensorreading or an inspection report, the computing device may use thereceived operational data as an input to the SCRBE model.

In some embodiments, the simulation tool 304 has generated a visualrendering of the composite model. In such embodiments, the simulationtool 304 may update the visual rendering of the composite model based onupdated output values generated by the composite model using thereceived first operational data.

The method 200 may include updating the composite model not just onceupon receipt of first operational data from a first operational datasource but many times upon receipt of a plurality of pieces ofoperational data from a plurality of operational data sources. Forexample, the digital twin may be constantly updated throughout theonline stage in order to keep “in sync” with the physical asset. Themethod 200 may execute steps for updating the model and any optionallygenerated visual rendering based upon receiving different data from thesame operational data source or based upon receiving different data froma different operational data source, or both. Therefore, the method 200may include receiving, by the computing device, from the firstoperational data source associated with the physical asset, secondoperational data associated with the at least one region of the physicalasset represented by the at least one parameter of the at least onecomponent in the plurality of components; updating, by the computingdevice, the composite model, based upon the received second operationaldata; and providing, by the computing device, a second recommendationfor maintaining the physical asset, based upon the updated compositemodel. In embodiments in which the simulation tool 304 has generated avisual rendering of the composite model, the simulation tool 304 mayupdate the visual rendering of the composite model based on updatedoutput values generated by the composite model using the received secondoperational data. Similarly, the method 200 may include receiving, bythe computing device, from a second operational data source associatedwith the physical asset, second operational data associated with atleast a second region of the physical asset represented by at least asecond parameter of at least a second component in the plurality ofcomponents; updating, by the computing device, the composite model,based upon the received second operational data; and providing, by thecomputing device, a second recommendation for maintaining the physicalasset, based upon the updated composite model. In embodiments in whichthe simulation tool 304 has generated a visual rendering of thecomposite model, the simulation tool 304 may update the visual renderingof the composite model based on updated output values generated by thecomposite model using the received second operational data.

As indicated above, the method 200 may include updating the Offlinestage periodically. Updating may include changing values within at leastone model in the plurality of models. Updating may include replacing atleast one model in the plurality of models to reflect new operatingconditions that have been observed (e.g., by operators generatinginspection reports or by sensors generating sensor data). Therefore, themethod 200 may include receiving second operational data from the firstoperational data source; updating at least one model in the plurality ofmodels based upon the received second operational data; analyzing, bythe computing device, for at least one model in the plurality of models,an error indicator identifying a level of error associated with the atleast one model, to determine whether the identified level of errorexceeds a tolerance level; increasing, by the computing device, a numberof basis functions in the port-reduced static condensation reduced basiselement approximation, based upon a determination that the at least onemodel has a level of error exceeding the tolerance level; repeating theanalyzing of the error indicator and the increasing of the number ofbasis functions for each model in the plurality of models until thelevel of error each of the plurality of models is beneath the tolerancelevel; updating, by the computing device, the composite model, basedupon the received data; and providing, by the computing device, arecommendation for maintaining the physical asset, based upon theupdated composite model. In embodiments in which the simulation tool 304has generated a visual rendering of the composite model, the simulationtool 304 may update the visual rendering of the composite model based onupdated output values generated by the composite model using thereceived second operational data. Similarly, the method 200 may includereceiving second operational data from a second operational data source;updating at least one model in the plurality of models based upon thereceived second operational; analyzing, by the computing device, for atleast one model in the plurality of models, an error indicatoridentifying a level of error associated with the at least one model, todetermine whether the identified level of error exceeds a tolerancelevel; increasing, by the computing device, a number of basis functionsin the port-reduced static condensation reduced basis elementapproximation, based upon a determination that the at least one modelhas a level of error exceeding the tolerance level; repeating, for eachmodel in the plurality of models, the analyzing of the error indicatorand the increasing of the number of basis functions until the level oferror each of the plurality of models is beneath the tolerance level;updating, by the computing device, the composite model, based upon thereceived data; and providing, by the computing device, a recommendationfor maintaining the physical asset, based upon the updated compositemodel. In embodiments in which the simulation tool 304 has generated avisual rendering of the composite model, the simulation tool 304 mayupdate the visual rendering of the composite model based on updatedoutput values generated by the composite model using the received secondoperational data.

The method 200 includes providing, by the computing device, arecommendation for maintaining the physical asset, based upon theupdated composite model (214). The system 100 may generate therecommendation based on at least one output value computed by theupdated composite model. Typical outputs of the digital thread areindustry-specific code checks for structural integrity, such asstrength, fatigue, and fitness-for-service standards from recognizedstandards bodies. The calculations required by these standards typicallyinvolve post-processing stress data (often based on hundreds orthousands of distinct load cases) in order to calculate quantities suchas remaining fatigue estimates or buckling utilization.

The online component 310 may generate the recommendation for maintainingthe physical asset. The online component 310 may provide therecommendation to the simulation tool 304 for display to the user viathe user interface 314. In addition to, or instead of, providingrecommendations for maintaining the physical asset, the computing device306 may provide a recommendation for identifying a plurality of aspectsof the physical asset to inspect the plurality of aspects rankedaccording to a level of priority, based upon the updated compositemodel. In addition to, or instead of, providing recommendations formaintaining the physical asset, the computing device 306 may provide arecommendation for determining a level of feasibility of a proposedmodification to the physical asset, based upon the updated compositemodel. In addition to, or instead of, providing recommendations formaintaining the physical asset, the computing device 306 may provide arecommendation for determining at least one operating condition of thephysical asset, based upon the updated composite model.

The user interface 314 may include a dashboard interface in which userscan click a button that will run the relevant standards-based checksbased on the current state of the digital twin (i.e. incorporating theup-to-date operational data) and generate a report based on the check.The report may include one or more recommendations, such as “everythingis OK”, or “There is an issue with XYZ region of the asset”, or “It isnot OK to operate the asset in Scenario XYZ”. This output can bepresented also as a “traffic light” for each asset, e.g. “green light”means all checks passed, “red light” means at least one check failed andindicates that there is something that needs further attention orinvestigation from the operator, who can then check the full report fordetails. In some embodiments, the system 100 includes functionality fortransmitting a report including the generated recommendations to anothercomputing device. In some embodiments, the system 100 includesfunctionality for implementing a recommendation.

Referring now to FIG. 2B, a block diagram depicts a visualization of themethod 200 resulting in a digital thread for a floating offshorestructure. Inputs are sensor data and inspection data, which are used toautomatically update and analyze the structural digital twin. Analysistypically consists of thousands of SCRBE solves in order to assessstrength and fatigue of the asset in its “as is” state. Once theanalysis is complete, reporting based on classification societystandards is automatically generated.

Referring now to FIG. 2C, a block diagram depicts a visualization of themethod 200 resulting in a digital thread for an offshore platform. Asshown in FIG. 2C, a plurality of accelerometers on a structure (e.g.,the physical asset) monitor the structural response to its environment.The method 200 executes and generates automated reporting based onindustry standards.

As has been shown, the SCRBE framework described above provides for thefour properties for digital twin modeling that were described above.Regarding holistic and detailed modeling, the SCRBE framework resolvesissues of costly computing, which, enables systems equivalent to 0(10⁷)or 0(10⁸) FE DOFs to be solved efficiently. This enables holistic anddetailed modeling of large-scale structural systems. Regarding speed,the SCRBE framework satisfies the requirement as a consequence of boththe component-local RB models and the port reduction; component-local RBenables fast modifications to a model, and port reduction enables fastsystem-level solves. Regarding parametric modeling, the component-localRB models introduce parametric ROM capabilities in a convenient andefficient manner, and this scales well to large numbers of parameterssince each per-component RB greedy is typically only required to dealwith a few parameters. Regarding standards compliance and certifiableaccuracy, SCRBE is fundamentally a physics-based method, which isformulated in terms of the same mesh-based approach as FE, except withacceleration due to projection onto a reduced set of port DOFs andcomponent-interior RB DOFs; this means that all of the standards forstructural integrity analysis that were typically designed for FE applydirectly to SCRBE models. Moreover, as described above, we can evaluateerror estimators or indicators for the SCRBE approach, which ensures theaccuracy of any solution that we compute in the Online stage. Hence, theSCRBE-based approach provides all of the key capabilities of digitaltwins in order to enable the structural digital twin workflows that areof interest for large-scale industrial systems.

The methods and systems described herein provide functionality forimplementing a SCRBE framework that provides powerful and uniquecapabilities for structural digital twins of large-scale assets. AnAdaptive ROM Enrichment methodology enables efficient and reliabletraining of SCRBE models in the Offline stage, as well as accuracyassessment and enrichment guidance (when needed) in the Online stage. Arange of structural analysis examples demonstrate the scalability,speed, and parametrization capabilities that are enabled by the SCRBEframework and the core concepts of a digital thread built aroundSCRBE-based structural digital twins were described above. The digitalthread described herein enables an automated framework that providesoperators with deeper structural integrity insights based on the “as is”state of critical assets, and hence empowers safer and more efficientoperations.

FIGS. 1B and 1C provide non-limiting examples of the methods and systemsdescribed above. Referring now to FIG. 1B, a block diagram depicts ahull model, updated based upon at least one value within an inspectionreport. The inspection report specifies a thickness for each entity(e.g., plate or stiffener) in the hull, so that a script canautomatically update the thicknesses in the corresponding entities inthe structural digital twin in order to incorporate the inspection data.Referring now to FIG. 1C, a block diagram depicts updated hull modelsused to generate automated buckling check reports. The figures on theleft show each entity in the hull (e.g., plates and stiffened panels).Stress data, shown in the middle-bottom figure, is extracted for eachentity and the buckling standards then specify a formula to evaluate foreach entity to provide a utilization value. In the middle-top figure weshow the utilization values as a “heat map” on the hull and any entitywith utilization exceeding 1 would be considered a failure. The systemthen generates a report (right) that identifies any entities that faileda buckling check. Such reports can be used by operators of the physicalasset for prioritizing inspection, maintenance, and repair, or assessingstructural health of the asset.

Referring now to FIGS. 4A, 4B, and 4C, block diagrams depict additionaldetail regarding computing devices that may be modified to executionfunctionality for implementing the methods and systems described above.Referring now to FIG. 4A, an embodiment of a network environment isdepicted. In brief overview, the network environment comprises one ormore clients 102 a-102 n (also generally referred to as local machine(s)102, client(s) 102, client node(s) 102, client machine(s) 102, clientcomputer(s) 102, client device(s) 102, computing device(s) 102,endpoint(s) 102, or endpoint node(s) 102) in communication with one ormore remote machines 106 a-106 n (also generally referred to asserver(s) 106 or computing device(s) 106) via one or more networks 404.

Although FIG. 4A shows a network 404 between the client(s) 102 and theremote machines 106, the client(s) 102 and the remote machines 106 maybe on the same network 404. The network 404 can be a local area network(LAN), such as a company Intranet, a metropolitan area network (MAN), ora wide area network (WAN), such as the Internet or the World Wide Web.In some embodiments, there are multiple networks 404 between theclient(s) and the remote machines 106. In one of these embodiments, anetwork 404′ (not shown) may be a private network and a network 404 maybe a public network. In another of these embodiments, a network 404 maybe a private network and a network 404′ a public network. In stillanother embodiment, networks 404 and 404′ may both be private networks.In yet another embodiment, networks 404 and 404′ may both be publicnetworks.

The network 404 may be any type and/or form of network and may includeany of the following: a point to point network, a broadcast network, awide area network, a local area network, a telecommunications network, adata communication network, a computer network, an ATM (AsynchronousTransfer Mode) network, a SONET (Synchronous Optical Network) network,an SDH (Synchronous Digital Hierarchy) network, a wireless network, anda wireline network. In some embodiments, the network 404 may comprise awireless link, such as an infrared channel or satellite band. Thetopology of the network 404 may be a bus, star, or ring networktopology. The network 404 may be of any such network topology as knownto those ordinarily skilled in the art capable of supporting theoperations described herein. The network 404 may comprise mobiletelephone networks utilizing any protocol or protocols used tocommunicate among mobile devices (including tables and handheld devicesgenerally), including AMPS, TDMA, CDMA, GSM, GPRS, UMTS, or LTE. In someembodiments, different types of data may be transmitted via differentprotocols. In other embodiments, the same types of data may betransmitted via different protocols.

A client(s) 102 and a remote machine 106 (referred to generally ascomputing devices 100) can be any workstation, desktop computer, laptopor notebook computer, server, portable computer, mobile telephone,mobile smartphone, or other portable telecommunication device, mediaplaying device, a gaming system, mobile computing device, or any othertype and/or form of computing, telecommunications or media device thatis capable of communicating on any type and form of network and that hassufficient processor power and memory capacity to perform the operationsdescribed herein. A client(s) 102 may execute, operate or otherwiseprovide an application, which can be any type and/or form of software,program, or executable instructions, including, without limitation, anytype and/or form of web browser, web-based client, client-serverapplication, an ActiveX control, or a JAVA applet, or any other typeand/or form of executable instructions capable of executing on client(s)102.

In one embodiment, a computing device 106 provides functionality of aweb server. In some embodiments, a web server 106 comprises anopen-source web server, such as the NGINX web servers provided by NGINX,Inc., of San Francisco, Calif., or the APACHE servers maintained by theApache Software Foundation of Delaware. In other embodiments, the webserver executes proprietary software, such as the INTERNET INFORMATIONSERVICES products provided by Microsoft Corporation of Redmond, Wash.,the ORACLE IPLANET web server products provided by Oracle Corporation ofRedwood Shores, Calif., or the BEA WEBLOGIC products provided by BEASystems of Santa Clara, Calif.

In some embodiments, the system may include multiple, logically-groupedremote machines 106. In one of these embodiments, the logical group ofremote machines may be referred to as a server farm 438. In another ofthese embodiments, the server farm 438 may be administered as a singleentity.

FIGS. 4B and 4C depict block diagrams of a computing device 100 usefulfor practicing an embodiment of the client(s) 102 or a remote machine106. As shown in FIGS. 4B and 4C, each computing device 100 includes acentral processing unit 421, and a main memory unit 422. As shown inFIG. 4B, a computing device 100 may include a storage device 428, aninstallation device 416, a network interface 418, an I/O controller 423,display devices 424 a-n, a keyboard 426, a pointing device 427, such asa mouse, and one or more other I/O devices 430 a-n. The storage device428 may include, without limitation, an operating system and software.As shown in FIG. 4C, each computing device 100 may also includeadditional optional elements, such as a memory port 403, a bridge 470,one or more input/output devices 430 a-n (generally referred to usingreference numeral 430), and a cache memory 440 in communication with thecentral processing unit 421.

The central processing unit 421 is any logic circuitry that responds toand processes instructions fetched from the main memory unit 422. Inmany embodiments, the central processing unit 421 is provided by amicroprocessor unit, such as: those manufactured by Intel Corporation ofMountain View, Calif.; those manufactured by Motorola Corporation ofSchaumburg, Ill.; those manufactured by Transmeta Corporation of SantaClara, Calif.; those manufactured by International Business Machines ofWhite Plains, N.Y.; or those manufactured by Advanced Micro Devices ofSunnyvale, Calif. Other examples include SPARC processors, ARMprocessors, processors used to build UNIX/LINUX “white” boxes, andprocessors for mobile devices. The computing device 400 may be based onany of these processors, or any other processor capable of operating asdescribed herein.

Main memory unit 422 may be one or more memory chips capable of storingdata and allowing any storage location to be directly accessed by themicroprocessor 421. The main memory 422 may be based on any availablememory chips capable of operating as described herein. In the embodimentshown in FIG. 4B, the processor 421 communicates with main memory 422via a system bus 450. FIG. 4C depicts an embodiment of a computingdevice 400 in which the processor communicates directly with main memory422 via a memory port 403. FIG. 4C also depicts an embodiment in whichthe main processor 321 communicates directly with cache memory 440 via asecondary bus, sometimes referred to as a backside bus. In otherembodiments, the main processor 421 communicates with cache memory 440using the system bus 450.

In the embodiment shown in FIG. 4B, the processor 421 communicates withvarious I/O devices 430 via a local system bus 450. Various buses may beused to connect the central processing unit 421 to any of the I/Odevices 430, including a VESA VL bus, an ISA bus, an EISA bus, aMicroChannel Architecture (MCA) bus, a PCI bus, a PCI-X bus, aPCI-Express bus, or a NuBus. For embodiments in which the I/O device isa video display 424, the processor 421 may use an Advanced Graphics Port(AGP) to communicate with the display 424. FIG. 4C depicts an embodimentof a computer 400 in which the main processor 421 also communicatesdirectly with an I/O device 430 b via, for example, HYPERTRANSPORT,RAPIDIO, or INFINIBAND communications technology.

One or more of a wide variety of I/O devices 430 a-n may be present inor connected to the computing device 400, each of which may be of thesame or different type and/or form. Input devices include keyboards,mice, trackpads, trackballs, microphones, scanners, cameras, and drawingtablets. Output devices include video displays, speakers, inkjetprinters, laser printers, 3D printers, and dye-sublimation printers. TheI/O devices may be controlled by an I/O controller 423 as shown in FIG.4B. Furthermore, an I/O device may also provide storage and/or aninstallation medium 416 for the computing device 400. In someembodiments, the computing device 400 may provide USB connections (notshown) to receive handheld USB storage devices such as the USB FlashDrive line of devices manufactured by Twintech Industry, Inc. of LosAlamitos, Calif.

Referring still to FIG. 4B, the computing device 100 may support anysuitable installation device 416, such as a floppy disk drive forreceiving floppy disks such as 3.5-inch, 5.25-inch disks or ZIP disks; aCD-ROM drive; a CD-R/RW drive; a DVD-ROM drive; tape drives of variousformats; a USB device; a hard-drive or any other device suitable forinstalling software and programs. In some embodiments, the computingdevice 400 may provide functionality for installing software over anetwork 404. The computing device 400 may further comprise a storagedevice, such as one or more hard disk drives or redundant arrays ofindependent disks, for storing an operating system and other software.Alternatively, the computing device 100 may rely on memory chips forstorage instead of hard disks.

Furthermore, the computing device 400 may include a network interface418 to interface to the network 404 through a variety of connectionsincluding, but not limited to, standard telephone lines, LAN or WANlinks (e.g., 802.11, T1, T3, 56kb, X.25, SNA, DECNET), broadbandconnections (e.g., ISDN, Frame Relay, ATM, Gigabit Ethernet,Ethernet-over-SONET), wireless connections, or some combination of anyor all of the above. Connections can be established using a variety ofcommunication protocols (e.g., TCP/IP, IPX, SPX, NetBIOS, Ethernet,ARCNET, SONET, SDH, Fiber Distributed Data Interface (FDDI), RS232, IEEE802.11, IEEE 802.11a, IEEE 802.11b, IEEE 802.11g, IEEE 802.11n,802.15.4, Bluetooth, ZIGBEE, CDMA, GSM, WiMax, and direct asynchronousconnections). In one embodiment, the computing device 400 communicateswith other computing devices 100′ via any type and/or form of gateway ortunneling protocol such as Secure Socket Layer (SSL) or Transport LayerSecurity (TLS). The network interface 418 may comprise a built-innetwork adapter, network interface card, PCMCIA network card, card busnetwork adapter, wireless network adapter, USB network adapter, modem,or any other device suitable for interfacing the computing device 100 toany type of network capable of communication and performing theoperations described herein.

In further embodiments, an I/O device 430 may be a bridge between thesystem bus 150 and an external communication bus, such as a USB bus, anApple Desktop Bus, an RS-232 serial connection, a SCSI bus, a FireWirebus, a FireWire 800 bus, an Ethernet bus, an AppleTalk bus, a GigabitEthernet bus, an Asynchronous Transfer Mode bus, a HIPPI bus, a SuperHIPPI bus, a SerialPlus bus, a SCI/LAMP bus, a FibreChannel bus, or aSerial Attached small computer system interface bus.

A computing device 400 of the sort depicted in FIGS. 4B and 4C typicallyoperates under the control of operating systems, which controlscheduling of tasks and access to system resources. The computing device400 can be running any operating system such as any of the versions ofthe MICROSOFT WINDOWS operating systems, the different releases of theUNIX and LINUX operating systems, any version of the MAC OS forMacintosh computers, any embedded operating system, any real-timeoperating system, any open source operating system, any proprietaryoperating system, any operating systems for mobile computing devices, orany other operating system capable of running on the computing deviceand performing the operations described herein. Typical operatingsystems include, but are not limited to: WINDOWS 3.x, WINDOWS 95,WINDOWS 98, WINDOWS 2000, WINDOWS NT 3.1-4.0, WINDOWS CE, WINDOWS XP,WINDOWS 7, WINDOWS 8, WINDOWS VISTA, and WINDOWS 10, all of which aremanufactured by Microsoft Corporation of Redmond, Wash.; any version ofMAC OS manufactured by Apple Inc. of Cupertino, Calif.; OS/2manufactured by International Business Machines of Armonk, N.Y.; Red HatEnterprise Linux, a Linus-variant operating system distributed by RedHat, Inc., of Raleigh, N.C.; Ubuntu, a freely-available operating systemdistributed by Canonical Ltd. of London, England; or any type and/orform of a Unix operating system, among others.

The computing device 400 can be any workstation, desktop computer,laptop or notebook computer, server, portable computer, mobile telephoneor other portable telecommunication device, media playing device, agaming system, mobile computing device, or any other type and/or form ofcomputing, telecommunications or media device that is capable ofcommunication and that has sufficient processor power and memorycapacity to perform the operations described herein. In someembodiments, the computing device 100 may have different processors,operating systems, and input devices consistent with the device. Inother embodiments, the computing device 400 is a mobile device, such asa JAVA-enabled cellular telephone/smartphone or personal digitalassistant (PDA). The computing device 400 may be a mobile device such asthose manufactured, by way of example and without limitation, by AppleInc. of Cupertino, Calif.; Google/Motorola Div. of Ft. Worth, Tex.;Kyocera of Kyoto, Japan; Samsung Electronics Co., Ltd. of Seoul, Korea;Nokia of Finland; Hewlett-Packard Development Company, L.P. and/or Palm,Inc. of Sunnyvale, Calif.; Sony Ericsson Mobile Communications AB ofLund, Sweden; or Research In Motion Limited of Waterloo, Ontario,Canada. In yet other embodiments, the computing device 100 is asmartphone, POCKET PC, POCKET PC PHONE, or other portable mobile devicesupporting Microsoft Windows Mobile Software.

In some embodiments, the computing device 400 is a digital audio player.In one of these embodiments, the computing device 400 is a digital audioplayer such as the Apple IPOD, IPOD TOUCH, IPOD NANO, and IPOD SHUFFLElines of devices manufactured by Apple Inc. In another of theseembodiments, the digital audio player may function as both a portablemedia player and as a mass storage device. In other embodiments, thecomputing device 100 is a digital audio player such as thosemanufactured by, for example, and without limitation, SamsungElectronics America of Ridgefield Park, N.J., or Creative TechnologiesLtd. of Singapore. In yet other embodiments, the computing device 400 isa portable media player or digital audio player supporting file formatsincluding, but not limited to, MP3, WAV, M4A/AAC, WMA Protected AAC,AEFF, Audible audiobook, Apple Lossless audio file formats, and .mov,.m4v, and .mp4 MPEG-4 (H.264/MPEG-4 AVC) video file formats.

In some embodiments, the computing device 400 includes a combination ofdevices, such as a mobile phone combined with a digital audio player orportable media player. In one of these embodiments, the computing device100 is a device in the Google/Motorola line of combination digital audioplayers and mobile phones. In another of these embodiments, thecomputing device 400 is a device in the IPHONE smartphone line ofdevices manufactured by Apple Inc. In still another of theseembodiments, the computing device 400 is a device executing the ANDROIDopen source mobile phone platform distributed by the Open HandsetAlliance; for example, the device 100 may be a device such as thoseprovided by Samsung Electronics of Seoul, Korea, or HTC Headquarters ofTaiwan, R.O.C. In other embodiments, the computing device 400 is atablet device such as, for example and without limitation, the IPAD lineof devices manufactured by Apple Inc.; the PLAYBOOK manufactured byResearch In Motion; the CRUZ line of devices manufactured by VelocityMicro, Inc. of Richmond, Va.; the FOLIO and THRIVE line of devicesmanufactured by Toshiba America Information Systems, Inc. of Irvine,Calif.; the GALAXY line of devices manufactured by Samsung; the HP SLATEline of devices manufactured by Hewlett-Packard; and the STREAK line ofdevices manufactured by Dell, Inc. of Round Rock, Tex.

A computing device 400 may be a file server, application server, webserver, proxy server, appliance, network appliance, gateway, applicationgateway, gateway server, virtualization server, deployment server, SSLVPN server, or firewall. In some embodiments, a computing device 400provides a remote authentication dial-in user service, and is referredto as a RADIUS server. In other embodiments, a computing device 100 mayhave the capacity to function as either an application server or as amaster application server. In still other embodiments, a computingdevice 400 is a blade server.

A computing device 400 may be referred to as a client node, a clientmachine, an endpoint node, or an endpoint. In some embodiments, a client400 has the capacity to function as both a client node seeking access toresources provided by a server and as a server node providing access tohosted resources for other clients.

In some embodiments, a first, client computing device 400 a communicateswith a second, server computing device 400 b. In one embodiment, theclient communicates with one of the computing devices 400 in a serverfarm. Over the network, the client can, for example, request executionof various applications hosted by the computing devices 400 in theserver farm and receive output data of the results of the applicationexecution for display.

Having described certain embodiments of methods and systems formaintaining a physical asset based on recommendations generated byanalyzing a model of the physical asset, the model comprising aplurality of components and forming a physics-based digital twin of thephysical asset, it will now become apparent to one of skill in the artthat other embodiments incorporating the concepts of the disclosure maybe used. Therefore, the disclosure should not be limited to certainembodiments, but rather should be limited only by the spirit and scopeof the following claims.

What is claimed is:
 1. A method for maintaining a physical asset basedon recommendations generated by analyzing a model of the physical asset,the model comprising a plurality of components and forming aphysics-based digital twin of the physical asset, the method comprising:(a) constructing, by a computing device, using a port-reduced staticcondensation reduced basis element approximation of at least a portionof a partial differential equation, a composite model of a plurality ofmodels, each of the plurality of models representing at least one of aplurality of components, each of the plurality of componentsrepresenting at least one region of a physical asset; (b) analyzing, bythe computing device, for at least one model in the plurality of models,an error indicator identifying a level of error associated with the atleast one model, to determine that the identified level of error exceedsa tolerance level; (c) increasing, by the computing device, a number ofbasis functions in the port-reduced static condensation reduced basiselement approximation, based upon a determination that the at least onemodel has a level of error exceeding the tolerance level; (d) repeating(b) and (c) for each model in the plurality of models until the level oferror for each of the plurality of models is beneath the tolerancelevel; (e) receiving, by the computing device, from a first operationaldata source associated with the physical asset, first operational dataassociated with at least one region of the physical asset represented byat least one parameter of at least one component in the plurality ofcomponents; (f) updating, by the computing device, the composite model,based upon the received first operational data; and (g) providing, bythe computing device, a recommendation for maintaining the physicalasset, based upon the updated composite model.
 2. The method of claim 1,wherein (a)-(d) are performed before (e).
 3. The method of claim 1,further comprising, after (a)-(d) and before (e), generating aphysics-based analysis of the physical asset using the composite model,wherein generating further comprises: receiving, by the computingdevice, first user input identifying an input value indicative of atleast one physical condition under which the physical asset is to beevaluated; and using, by the computing device, the composite model togenerate at least one output value based at least in part on the atleast one input value, wherein the at least one output value isindicative of a behavior of the physical system under the at least onephysical condition, wherein the at least one output value comprises aplurality of output values over an N-dimensional domain.
 4. The methodof claim 3, wherein receiving further comprises receiving, by thecomputing device, user input identifying an input value extracted froman inspection report based on a physical inspection of the physicalasset.
 5. The method of claim 3, wherein receiving further comprisesreceiving, by the computing device, user input identifying an inputvalue extracted from operational data received from a sensor associatedwith the physical asset.
 6. The method of claim 1 further comprising:(h) generating, by a simulation tool executed by the computing device, avisual rendering of the composite model including a visualization of atleast one result of a physics-based analysis of the physical asset. 7.The method of claim 6 further comprising: (i) updating, by thesimulation tool, the visual rendering, based upon the received firstoperational data.
 8. The method of claim 1, wherein (d) comprisesrepeating (b) and (c) for each model in the plurality of models until athreshold number of iterations is reached.
 9. The method of claim 1,wherein (e) further comprises receiving, by the computing device, fromthe first operational data source associated with the physical asset,first operational data generated by a sensor associated with thephysical asset.
 10. The method of claim 1, wherein (e) further comprisesreceiving, by the computing device, from the first operational datasource associated with the physical asset, first operational dataextracted from an inspection report associated with the physical asset.11. The method of claim 1, wherein (e) further comprises receiving, bythe computing device, from the first operational data source associatedwith the physical asset, first operational data extracted from a reportgenerated by an operator of the physical asset.
 12. The method of claim1 further comprising: (h) providing, by the computing device, arecommendation for identifying a plurality of aspects of the physicalasset to inspect the plurality of aspects ranked according to a level ofpriority, based upon the updated composite model.
 13. The method ofclaim 1 further comprising: (h) providing, by the computing device, arecommendation for determining a level of feasibility of a proposedmodification to the physical asset, based upon the updated compositemodel.
 14. The method of claim 1 further comprising: (h) providing, bythe computing device, a recommendation for determining a level ofoperability of the physical asset, based upon the updated compositemodel.
 15. The method of claim 1 further comprising: (h) receiving, bythe computing device, from the first operational data source associatedwith the physical asset, second operational data associated with the atleast one region of the physical asset represented by the at least oneparameter of the at least one component in the plurality of components;(i) updating, by the computing device, the composite model, based uponthe received second operational data; and (j) providing, by thecomputing device, a second recommendation for maintaining the physicalasset, based upon the updated composite model.
 16. The method of claim 1further comprising: (h) receiving, by the computing device, from asecond operational data source associated with the physical asset,second operational data associated with at least a second region of thephysical asset represented by at least a second parameter of at least asecond component in the plurality of components; (i) updating, by thecomputing device, the composite model, based upon the received secondoperational data; and (j) providing, by the computing device, a secondrecommendation for maintaining the physical asset, based upon theupdated composite model.
 17. The method of claim 1 further comprising:(h) receiving second operational data from the first operational datasource; (i) updating at least one model in the plurality of models basedupon the received second operational data; (j) analyzing, by thecomputing device, for at least one model in the plurality of models, anerror indicator identifying a level of error associated with the atleast one model, to determine whether the identified level of errorexceeds a tolerance level; (k) increasing, by the computing device, anumber of basis functions in the port-reduced static condensationreduced basis element approximation, based upon a determination that theat least one model has a level of error exceeding the tolerance level;(l) repeating (b) and (c) for each model in the plurality of modelsuntil the level of error each of the plurality of models is beneath thetolerance level; (m) updating, by the computing device, the compositemodel, based upon the received second operational data; and (n)providing, by the computing device, a recommendation for maintaining thephysical asset, based upon the updated composite model.
 18. The methodof claim 1 further comprising: (h) receiving second operational datafrom a second operational data source; (i) updating at least model inthe plurality of models based upon the received second operational data;(j) analyzing, by the computing device, for at least one model in theplurality of models, an error indicator identifying a level of errorassociated with the at least one model, to determine whether theidentified level of error exceeds a tolerance level; (k) increasing, bythe computing device, a number of basis functions in the port-reducedstatic condensation reduced basis element approximation, based upon adetermination that the at least one model has a level of error exceedingthe tolerance level; (l) repeating (b) and (c) for each model in theplurality of models until the level of error each of the plurality ofmodels is beneath the tolerance level; (m) updating, by the computingdevice, the composite model, based upon the received second operationaldata; and (n) providing, by the computing device, a recommendation formaintaining the physical asset, based upon the updated composite model.19. A non-transitory, computer-readable medium encoded withcomputer-executable instructions that, when executed on a computingdevice, cause the computing device to carry out a method for maintaininga physical asset based on recommendations generated by analyzing a modelof the physical asset, the model comprising a plurality of componentsand forming a physics-based digital twin of the physical asset, themethod comprising: (a) constructing, by a computing device, using aport-reduced static condensation reduced basis element approximation ofat least a portion of a partial differential equation, a composite modelof a plurality of models, each of the plurality of models representingat least one of a plurality of components, each of the plurality ofcomponents representing at least one region of a physical asset (b)analyzing, by the computing device, for at least one model in theplurality of models, an error indicator identifying a level of errorassociated with the at least one model, to determine that the identifiedlevel of error exceeds a tolerance level; (c) increasing, by thecomputing device, a number of basis functions in the port-reduced staticcondensation reduced basis element approximation, based upon adetermination that the at least one model has a level of error exceedingthe tolerance level; (d) repeating (b) and (c) for each model in theplurality of models until the level of error for each of the pluralityof models is beneath the tolerance level; (e) receiving, by thecomputing device, from a first operational data source associated withthe physical asset, first operational data associated with at least oneregion of the physical asset represented by at least one parameter of atleast one component in the plurality of components; updating, by thecomputing device, the composite model, based upon the received firstoperational data; and (g) providing, by the computing device, arecommendation for maintaining the physical asset, based upon theupdated composite model.